C₂×C₃×#462 #84897

order	384
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

480 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 4 generators on 25 points:

(1,2)(3,4,5);(7,9)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19);(1,2)(6,7,8,9);(1,2)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 17473);